Difference between revisions of "Exponential function"
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The '''exponential function''' is the [[function]] <math>f(x) = e^x</math>, [[exponentiation]] by ''[[e]]''. It is a very important function in [[analysis]], both [[real]] and [[complex]]. | The '''exponential function''' is the [[function]] <math>f(x) = e^x</math>, [[exponentiation]] by ''[[e]]''. It is a very important function in [[analysis]], both [[real]] and [[complex]]. | ||
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== General Info and Definitions == | == General Info and Definitions == | ||
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Exponential functions are functions that grows or decays at a constant percent rate. | Exponential functions are functions that grows or decays at a constant percent rate. | ||
:Exponential functions that result in an '''''increase''''' of ''y'' is called an '''''exponential growth'''''. | :Exponential functions that result in an '''''increase''''' of ''y'' is called an '''''exponential growth'''''. | ||
:Exponential functions that result in an '''''decrease''''' of ''y'' is called an '''''exponential decay'''''. | :Exponential functions that result in an '''''decrease''''' of ''y'' is called an '''''exponential decay'''''. | ||
| + | An exponential growth graph looks like: | ||
| − | + | [[Image:2_power_x_growth.jpg]] | |
| − | An exponential decay graph looks like: | + | An exponential decay graph looks like: |
| + | [[Image:05_power_x_decay.jpg]] | ||
Exponential functions are in one of three forms. | Exponential functions are in one of three forms. | ||
:<math>f\left( x \right) = ab^x </math>, where ''b'' is the % change written in decimals | :<math>f\left( x \right) = ab^x </math>, where ''b'' is the % change written in decimals | ||
| − | :<math>f\left( x \right) = ae^k </math>, where | + | :<math>f\left( x \right) = ae^k </math>, where [[e]] is the irrational constant ''2.71828182846....'' |
:<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}} | :<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}} | ||
</math> or <math>f\left( x \right) = a\left( 2 \right)^{{x \over d}} | </math> or <math>f\left( x \right) = a\left( 2 \right)^{{x \over d}} | ||
</math>, where ''h'' is the half-life (for decay), or ''d'' is the doubling time (for growth). | </math>, where ''h'' is the half-life (for decay), or ''d'' is the doubling time (for growth). | ||
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Whether an exponential function shows growth or decay depends upon the value of its ''b'' value. | Whether an exponential function shows growth or decay depends upon the value of its ''b'' value. | ||
| − | :If <math>b > 1</math>, then the | + | :If <math>b > 1</math>, then the function will show growth. |
| − | :If <math>0 < b < 1</math>, then the function will show decay. | + | :If <math>0 < b < 1</math>, then the function will show decay. |
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== Solving Exponential Equations == | == Solving Exponential Equations == | ||
| − | + | There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using [[logarithms]]. | |
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| − | There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using logarithms. | ||
'''Example:''' Solve <math>56 = 12\left( {1.24976} \right)^x </math> | '''Example:''' Solve <math>56 = 12\left( {1.24976} \right)^x </math> | ||
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::[[Image:expfunc_graphsolve_eqn.jpg]] | ::[[Image:expfunc_graphsolve_eqn.jpg]] | ||
| − | *''' | + | *'''Algebraically:''' |
| + | There, we will use [[Natural logarithm|natural logarithms]]. The same operation can also be done with [[common logarithms]]. | ||
::<math>56 = 12\left( {1.24976} \right)^x </math> | ::<math>56 = 12\left( {1.24976} \right)^x </math> | ||
::<math>{{56} \over {12}} = \left( {1.24976} \right)^x </math> | ::<math>{{56} \over {12}} = \left( {1.24976} \right)^x </math> | ||
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::<math>x = {{\ln \left( {{{56} \over {12}}} \right)} \over {\ln \left( {1.24976} \right)}}</math> | ::<math>x = {{\ln \left( {{{56} \over {12}}} \right)} \over {\ln \left( {1.24976} \right)}}</math> | ||
::<math>x \approx 6.9093</math> | ::<math>x \approx 6.9093</math> | ||
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Latest revision as of 14:57, 6 March 2022
The exponential function is the function 
, exponentiation by e. It is a very important function in analysis, both real and complex.
General Info and Definitions
Exponential functions are functions that grows or decays at a constant percent rate.
- Exponential functions that result in an increase of y is called an exponential growth.
- Exponential functions that result in an decrease of y is called an exponential decay.
An exponential growth graph looks like:
An exponential decay graph looks like:
Exponential functions are in one of three forms.
, where b is the % change written in decimals
, where e is the irrational constant 2.71828182846....
or 
, where h is the half-life (for decay), or d is the doubling time (for growth).
, where b is the % change written in decimals
, where 
or 
, where h is the half-life (for decay), or d is the doubling time (for growth).Whether an exponential function shows growth or decay depends upon the value of its b value.
- If
, then the function will show growth. - If
, then the function will show decay.
, then the function will show growth.
, then the function will show decay.Solving Exponential Equations
There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using logarithms.
Example: Solve 
- Graphically:
- Graph both equations and find the intersection.
- Algebraically:
There, we will use natural logarithms. The same operation can also be done with common logarithms.
